630 
PROFESSOR SYLVESTER ON THE REAL 
round the axis of D the facultative and non-facultative portions may be made to exchange 
places. 
(46) The axis of D itself lies on the surface of G, and like every other portion of this 
surface is facultative, for there is no reason for disallowing G to become zero. Con- 
versely, if, instead of a real equation, we take one of the conjugate class (described in 
the second section), the whole of the facultative portion of space (except the separating 
surface G) becomes non-facultative, and the non-facultative part becomes facultative, 
hut G itself remains facultative. When the invariants, or any of them, become imagi- 
nary, we are put out of space altogether, and the system can belong neither to a real 
nor to a conjugate family, but to one with coefficients at the same time imaginary and 
non-conjugate. G=Q( 43 ), it may be remarked, will in all cases be the condition of 
an equation capable of linear transformation into one of recurrent ( 44 ) form ; for the 
reduced form then in general becomes ru h -\- rv 5 — t(u-\- v) 5 . The case when G becomes 
zero by virtue of J=0 and L=0, that is to say when the function is reducible by real 
or imaginary linear substitutions (see footnote ( 39 ) ( f )) to the form u(u 4 +v 4 ), is the 
one which might for a moment be supposed to offer an exception to the rule ; but only 
the exception is only apparent, since u[u' 4 — v 4 ), on writing u—p- [-q, v=jq — q, becomes 
1 6 (j?+M/+ 2 s ). 
(47) To every point in space, it has been remarked, will correspond one particular 
family of equations all of the same character as regards the number they contain of 
real or imaginary roots, because capable of being derived from one another by real 
linear substitutions, such family consisting of an infinite number of ordinary or con- 
jugate equations according as the point is facultative or non-facultative; but it may be 
well to notice that, conversely, every point does not correspond to a distinct family. In 
fact every point in the curves D=pJ 2 , ~L=qJ 3 (y>, q being constants) will denote a curve 
divided into two branches by the origin of coordinates, one of which will be facultative 
and the other non-facultative ; but in each separate branch every point will represent 
the very same family. Any such separate branch may be termed an isomorphic line ; 
and we see that the whole of space may he conceived as permeated by and made up of 
such lines radiating out from the origin in all directions. 
(48) The origin at which J = 0, D=p 0, L=0, as already noticed, corresponds to the 
case of three equal roots. The theorem that, when more than half as many roots are 
equal to each other as there are units in the degree of any binary form, all the inva- 
riants vanish, was remarked by myself originally in the very infancy of the subject, 
before Mr. Cayley’s paper, alluded to by M. TIermite, appeared in Crelle. The method 
of proof which then occurred to me is the simplest that can be given. For instance, in 
( 43 ) I shall hereafter allude to the surface denoted by G=0 under the name of the Amphigenous Surface, 
as being the locus of the points which give birth to real and conjugate forms indifferently. 
( 44 ) The roots of recurring equations, geometrically represented, in general go in quadruplets, A, A' ; B, B f , 
where A and B, as also A', B', are mutual optical images of each other in respect to a fixed line, and A, A', as 
also B, B', are electrical images of each other in respect to a circle of which the fixed line is a diameter — with 
liberty, of course, for the images taken in either mode of combination to coalesce so as to reduce the quadruplet 
to a simple pair. 
